lecture1

lecture1 - Lecture 1 Notes: 06 / 27 The first part of this...

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Lecture 1 Notes: 06 / 27 The first part of this class will primarily cover oscillating systems (harmonic oscillators and waves). These systems are very common in nature - a system displaced from equilibrium by a small amount will tend to oscillate harmonically, unless the friction is too high for oscillations to take place. Mass on a spring and Hooke's Law Simple system: Mass on a spring. Consider a spring that has a relaxed length L , attached to a mass on a frictionless table: Let x be the distance by which the spring is stretched or compressed from its relaxed length. It's positive if the spring is stretched, and negative if it is compressed. If x is sufficiently small (compared to L ), we find that the force is proportional to the displacement: F = -k x This is Hooke's law. The sign is negative because if the mass is displaced to the right, the force tries to return it back to the left, towards the equilibrium position, and vice versa. For this reason, the force is called the restoring force . The proportionality constant is called the spring constant , and measures the stiffness of the spring. Potential energy of the spring: Since F = -dU / dx , the potential energy is The total energy is the kinetic energy plus the potential energy: Systems that obey Hooke's Law are called harmonic oscillators .
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Why Hooke's Law? Why does the spring (and many other systems in nature) follow such a simple force law? There is a mathematical reason. In general, the force F is a complicated function of the displacement x . At the equilibrium position, x = 0, the force is zero. For small displacements from equilibrium, we can fit any F with a polynomial in x (this is known as a Taylor series, you should have seen it in your calculus class.) F = A + Bx + Cx 2 +Dx 3 + . .. Since F(0) = 0 (equilibrium), A = 0. Therefore, F = Bx + Cx 2 +Dx 3 + . .. Now, assume that the displacement
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lecture1 - Lecture 1 Notes: 06 / 27 The first part of this...

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